Structural Measures as Guides to Ultrastable States in Overjammed Packings

Jammed, disordered packings of given sets of particles possess a multitude of equilibrium states with different mechanical properties. Identifying and constructing desired states, e.g., of superior stability, is a complex task. Here, we show that in two-dimensional particle packings the energy of all metastable states (inherent structures) is reliably classified by simple scalar measures of local steric packing. These structural measures are insensitive to the particle interaction potential and so robust that they can be used to guide a modified swap algorithm that anneals polydisperse packings toward low-energy metastable states exceptionally fast. The low-energy states are extraordinarily stable against applied shear, so that the approach also efficiently identifies ultrastable packings.

Figure 1

(a) Schematic energy landscape of disordered systems and two example packing metastable states of high energy (left) and low energy (right). Structural measures of metastable particle configurations: (b) contact number (color coded); (c) gap space; (d) radical tessellation to compute equivalent foam energy; and (e) local steric angles for computation of the angle order parameter.

Figure 2

(a) Example metastable states for the monodisperse particle system, $N=400$, metastable energy increases from left to right. Each particle is color coded by contact number. Structural measures correlating with MS energy: (b) contact number, (c) gap space, (d) equivalent foam energy, and (e) angle order parameter. Gray symbols are individual MS ($n=5000$ samples), red symbols are binned mean and standard deviation.

Figure 3

(a) Example metastable states for the polydisperse particle system, $c_A=0.4$ and $N=400$, metastable energy increases from left to right. Each particle is color coded by contact number. Structural measures correlating with MS energy: (b) contact number, (c) gap space, (d) equivalent foam energy, and (e) angle order parameter. Gray points result from annealing different initial configurations to the nearest MS ($n=9000$ samples), and magenta points are obtained by the Monte Carlo swap algorithm ($n=173000$). Blue symbols are binned mean and standard deviation.

Figure 4

(a) Schematic showing a single area swap between particles $i$ and $j$. (b) A simultaneous area swap algorithm to minimize the angle order parameter, demonstrated for $N=144$. Individual particles are color coded in terms of their ${\mathrm{\Theta }}_i$ values. (c) Energy ${\epsilon }_r$, (d) mean number of contacts $z$, and (e) angle order parameter $\mathrm{\Theta }$ of MS obtained by the angle swap algorithm for different system sizes, average over 1000 MS for each $N$. Green and red data points represent initial and final MS, respectively. (f) Angle swap simulations reduce energy far faster than Monte Carlo swap simulations. Mean and standard deviation of energies of 50 initially random packings are shown against number of FIRE energy minimizations. Blue: MC swap; the gray dashed line indicates the MC mean energy after ${10}^5$ steps. Red: AS algorithm, reaching this energy after 40 steps.

Figure 5

(a) Schematics of pure shear simulations. (b) Examples of energy as a function of strain starting at high (cyan) and low (brown) energy. Strain values ${\gamma }_c$ for the first irreversible rearrangement are indicated by arrows. (c) Critical strain ${\gamma }_c$ as a function of state energy ${\epsilon }_r$. (d) Energy difference between final MS and initial MS induced by irreversible rearrangement. Gray symbols are individual MS ($n=450$), colored symbols are binned mean and standard deviation.